Slice

For [math]\displaystyle{ n\in{\Bbb Z}_+ }[/math] and [math]\displaystyle{ (a,b,c) }[/math] in the triangular grid

[math]\displaystyle{ \Delta_n := \{ (a,b,c) \in {\Bbb Z}_+^3: a+b+c = n \}, }[/math]

the slice [math]\displaystyle{ \Gamma_{a,b,c} }[/math] is the set of all strings in [math]\displaystyle{ [3]^n }[/math] with [math]\displaystyle{ a }[/math] occurences of [math]\displaystyle{ 1 }[/math], [math]\displaystyle{ b }[/math] occurences of [math]\displaystyle{ 2 }[/math], and [math]\displaystyle{ c }[/math] occurences of [math]\displaystyle{ 3 }[/math]. Thus, for instance,

[math]\displaystyle{ \Gamma_{1,2,0} = \{ 011, 101, 101 \}. }[/math]

The cube [math]\displaystyle{ [3]^n }[/math] has [math]\displaystyle{ \frac{(n+1)(n+2)}{2} }[/math] slices, and each slice [math]\displaystyle{ \Gamma_{a,b,c} }[/math] has cardinality [math]\displaystyle{ \frac{n!}{a!b!c!} }[/math].

The slice [math]\displaystyle{ \Gamma_{a,b,c} }[/math] is incident to [math]\displaystyle{ 2^a+2^b+2^c-3 }[/math] combinatorial lines.

The equal-slices measure on [math]\displaystyle{ [3]^n }[/math] gives each slice a total measure of [math]\displaystyle{ 1 }[/math].

The [math]\displaystyle{ k=2 }[/math] analogue of a slice, sometimes called a layer, plays a role in the proof of Sperner's theorem.

A combinatorial line must touch three slices [math]\displaystyle{ \Gamma_{a+r,b,c}, \Gamma_{a,b+r,c}, \Gamma_{a,b,c+r} }[/math] corresponding to an equilateral triangle in [math]\displaystyle{ \Delta_n }[/math]. Conversely, one might hope that any sufficiently "rich" subsets of three slices [math]\displaystyle{ \Gamma_{a+r,b,c}, \Gamma_{a,b+r,c}, \Gamma_{a,b,c+r} }[/math] in an equilateral triangle will have many combinatorial lines between them. However there appear to be quite a lot of obstructions to this hope, consider e.g. the subset of [math]\displaystyle{ \Gamma_{a+r,b,c} }[/math] of strings whose first digit is [math]\displaystyle{ 1 }[/math], and the subsets of [math]\displaystyle{ \Gamma_{a,b+r,c}, \Gamma_{a,b,c+r} }[/math] of strings whose first digit is [math]\displaystyle{ 2 }[/math].